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arXiv:cond-mat/9801179 · Physical Review Letters

Small Superconducting Grain in the Canonical Ensemble

Andrea Mastellone, G. Falci, Rosario Fazio

Published May 18, 1998• 137 citations
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Abstract

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground-state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing $\ensuremath{\delta}$. Both quantities turn out to be parity dependent and universal functions of the ratio $\ensuremath{\delta}/\ensuremath{\Delta}$ ( $\ensuremath{\Delta}$ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.

DOI
10.1103/physrevlett.80.4542

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By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground-state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing \ensuremathδ\ensuremath{\delta}\ensuremathδ. Both quantities turn out to be parity dependent and universal functions of the ratio \ensuremathδ/\ensuremathΔ\ensuremath{\delta}/\ensuremath{\Delta}\ensuremathδ/\ensuremathΔ ( \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.
The canonical BCS wave function is tested for the attractive Hubbard model. Results are presented for one dimension, and are compared with the exact solutions by the Bethe ansatz and the results from the conventional grand canonical BCS approximation, for various chain lengths, electron densities, and coupling strengths. While the exact ground state energies are reproduced very well both by the canonical and grand canonical BCS approximations, the canonical method significantly improves the energy gaps for small systems and weak coupling. The ``parity'' effect emerges naturally in our canonical results, as a positive gap for an even number of electrons and a negative ``gap'' for an odd number of electrons. Furthermore, we find a ``super-even'' effect: the energy gap oscillates as a function of even electron number, depending on whether the number of electrons is 4m4m4m or 4m+2(m4m+2 (m4m+2(m integer). Such oscillations as a function of electron number should be observable with tunneling measurements in ultrasmall metallic grains.
We study the properties of ultrasmall metallic grains with sizes in the range 20--400 electrons. Using a particle-hole version of the density-matrix renormalization-group (DMRG) method we compute condensation energies, spectroscopic gaps, pairing parameters, and particle-hole probabilities of the ground-state wave function. The results presented in this paper confirm that the bulk superconducting regime (large grains) and the fluctuation dominated regime (small grains) are qualitatively different, but show that the crossover between them is very smooth with no signs of critical level spacings separating them. We compare our DMRG results with the exact ones obtained with the Richardson solution finding complete agreement. We also propose a simplified version of the DMRG wave function, called the particle-hole BCS Ansatz, which agrees qualitatively with the DMRG solution and illustrates what is lacking in the projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.
We test the canonical BCS wave functions for fixed number of electrons for the attractive Hubbard model. We present results in one dimension for various chain lengths, electron densities, and coupling strengths. The ground-state energy and energy gap to the first excited state are compared with the exact solutions obtained by the Bethe ansatz as well as with the results from the conventional grand-canonical BCS approximations. While the canonical and grand-canonical BCS results are both in very good agreement with the exact results for the ground state energies, improvements due to conserving the electron number in finite systems are manifest in the energy gap. As the system size is increased, the canonical results converge to the grand-canonical ones. The ``parity'' effect that arises from the number parity (even or odd) of electrons are studied with our canonical scheme.